Question: Find $\dfrac{d}{dx}(3\cdot5^x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $5^x\cdot 3\ln(3)$ (Choice B) B $5^x\cdot 3\ln(5)$ (Choice C) C $3\cdot 5^{x}$ (Choice D) D $3\cdot 5^{x-1}$
Solution: The expression to differentiate includes an exponential term. Remember that the derivative of the general exponential term $a^x$ (where $a$ is any positive constant) is $\ln(a)\cdot a^x$. Put another way, $\dfrac{d}{dx}(a^x)=\ln(a)\cdot a^x$. $\begin{aligned} &\phantom{=}\dfrac{d}{dx}(3\cdot5^x) \\\\ &=3\dfrac{d}{dx}(5^x) \\\\ &=3\cdot\ln(5)\cdot5^x \\\\ &=5^x\cdot 3\ln(5) \end{aligned}$ In conclusion, $\dfrac{d}{dx}(3\cdot5^x)=5^x\cdot 3\ln(5)$.